| import OAI.Probability.ContinuumTransition.UniformMin |
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| namespace OAI |
|
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| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
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| def moderateBox : Set Parameter := {x | 1/4 ≤ x.1 ∧ x.1 ≤ 2 ∧ x.2 ≤ 10} |
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| lemma initial_ball_moderate : closedBall initialPoint (1/2) ⊆ moderateBox := by |
| intro x hx |
| have hh : |x.1-1| ≤ 1/2 ∧ |x.2-(-3/4)| ≤ 1/2 := by |
| simpa only [mem_closedBall,dist_eq_norm_sub,Prod.norm_def,Prod.fst_sub,Prod.snd_sub, |
| Real.norm_eq_abs,max_le_iff,initialPoint] using hx |
| obtain ⟨h1,h2⟩ := hh |
| obtain ⟨h1,h1'⟩ := abs_le.mp h1 |
| obtain ⟨h2,h2'⟩ := abs_le.mp h2 |
| exact ⟨by linarith,by linarith,by linarith⟩ |
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| lemma moderateBox_subset : moderateBox ⊆ pressureDomain := by |
| intro x hx |
| change 0 < x.1 |
| linarith only [hx.1] |
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| lemma physicalProfile_zero (t : ℝ) (R : ℕ → ℝ) : |
| physicalProfile t (fun _ => 0) R = coreProfile t := by |
| funext r |
| simp [physicalProfile,attraction] |
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| lemma physical_convex_bounded {t : ℝ} (ht : 1 < t) {σ R : ℕ → ℝ} |
| (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hm : 17576*kernelHeight*(∑' n, σ n) ≤ t) : |
| ConvexOn ℝ pressureDomain (physicalPressure (physicalProfile t σ R) t) := by |
| have ht0 := lt_trans zero_lt_one ht |
| exact convexOn_physicalPressure (measurable_radialPotential (measurable_physicalProfile t hσ hR hs)) ht |
| (physicalProfile_stable ht0.le hσ hR hs (strengthened_smallness ht0 hm).1) |
| (fun r _ => physicalProfile_le_core ht0.le hσ hR r) |
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| lemma physical_meanField_bounded {t a : ℝ} (ht : 1 < t) (ha : 0 < a) {σ R : ℕ → ℝ} |
| (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hm : 17576*kernelHeight*(∑' n, σ n) ≤ t) (hat : 1024*a < t) : |
| IsMeanField (physicalPressure (physicalProfile t σ R) t) a |
| (meanField (physicalPressure (physicalProfile t σ R) t) a) ∧ |
| (∀ x ∈ pressureDomain, ∃ d, quadraticCost (physicalPressure (physicalProfile t σ R) t) a x d = |
| meanField (physicalPressure (physicalProfile t σ R) t) a x) := by |
| have ht0 := lt_trans zero_lt_one ht |
| have hsmall := (strengthened_smallness ht0 hm).1 |
| have hgap := strengthened_gap ht0 hm hat |
| exact ⟨isMeanField_meanField (physical_meanField ht ha.le hσ hR hs hsmall hgap.le), |
| fun _ hx => physical_meanField_attained ht ha hσ hR hs hsmall hgap hx⟩ |
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| lemma physical_meanField_bounded_attainment {t a : ℝ} (ht : 1 < t) (ha : 0 < a) {σ R : ℕ → ℝ} |
| (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hm : 17576*kernelHeight*(∑' n, σ n) ≤ t) (hat : 1024*a < t) |
| {x : Parameter} (hx : x ∈ moderateBox) : |
| ∃ d ∈ Icc (0:ℝ) 1048576, quadraticCost (physicalPressure (physicalProfile t σ R) t) a x d = |
| meanField (physicalPressure (physicalProfile t σ R) t) a x := by |
| have hh := physical_meanField_bounded ht ha hσ hR hs hm hat |
| obtain ⟨w,hw⟩ := activeLifts_nonempty ha (physical_convex_bounded ht hσ hR hs hm) hh.1 |
| (moderateBox_subset hx) (hh.2 _ (moderateBox_subset hx)) |
| exact ⟨w.2,physical_transform_active_bound ht ha hσ hR hs hm hat hx.1 hx.2.1 hx.2.2 hw,hw.1⟩ |
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| lemma strong_smallness {t m : ℝ} (ht : 0 < t) |
| (hm : 140608*kernelHeight*m ≤ t) : 2197*t*kernelHeight*m ≤ (t^2/8)/8 := by |
| have h := mul_le_mul_of_nonneg_left hm ht.le |
| nlinarith only [h] |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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